Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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17
votes
5answers
844 views

A strange combinatorial identity [duplicate]

In reading about A polarization identity for multilinear maps by Erik G F Thomas, I am led to prove the following combinatorial identity, which I cannot find anywhere, nor do I have any idea how to ...
0
votes
0answers
30 views

Proving primality of $p$ without making any calculation involving $p$ directly

Wilson's Theorem states that a positive integer $p > 1$ is prime if and only if $(p-1)! \equiv -1 \pmod p$, showing a relationship between factorials and prime numbers. Finding it curious, today I ...
-3
votes
0answers
44 views

Evaulate the expression [on hold]

$$x^2 + \frac53x^3 + \frac{23}{12}x^4 + \frac{119}{60}x^5 + \dots + 2\times \frac{5000!-1}{5000!}x^{5000}$$ How to evaluate the expression for $x=0.7893$ to the nearest $20$ decimal places?
1
vote
0answers
24 views

multifactorial of non-integer

I want to calculate 12.1!!!!!! , Just for curiosity. (One of my friend texted the term to me for some complex reason.) I searched for multifactorial in terms of gamma function or equation, and found ...
0
votes
0answers
19 views

A new formula relating the factorial and Riemann Zeta function resp. Bernoulli numbers?

I proved the following identities involving the factorial and Riemann's Zeta function respectively the Bernoulli numbers: $$\sum _{k=1}^{i}-{\frac {{\pi }^{-2\,k}\zeta \left( 2\,k \right)\left( -1 ...
0
votes
2answers
35 views

Mathematical Induction with series and factorials.

I wish to show the following $$ a_{n}=\sum_{k=0}^{n}\frac{1}{(2k+1)!(2(n-k)+1)!}=\sum_{k=0}^{n+1}\frac{1}{(2k)!(2(n+1-k))!}=b_{n+1} $$ for $n\geq0$ and wish to do it using induction. I've shown it ...
2
votes
2answers
48 views

Factoring the factorials

Just for the fun of it, I've started factoring $n!$ into its prime divisors, and this is what I got for $2\leq n\leq20$: $$\begin{align} 2! &= 2^\color{red}{1} &S_e=1\\ 3! &= ...
1
vote
3answers
702 views

How to prove the sum of combination is equal to $2^n - 1$

One of the algorithm I learnt involve these steps: $1$. define a set $S$ of $n$ elements $2$. form a subset $S'$ of $k$ choice from $n$ elements of the set $S$ ($k$ starts with $1$), which is ...
3
votes
0answers
105 views

Solving equation involving factorials

I have this particular equation $\frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!} = \frac{\Gamma(p)(1+q)^{n+2p} 2^n}{q^{p}(2+q)^{n+p}}$. Now, given the values of $\alpha$ and $\beta$, I need to find ...
2
votes
2answers
61 views

Factorial Divisibility

Let $a$ and $b$ be positive integers greater than one. With that in mind, $$(a \cdot b)!$$ is not necessarily divisible by: a) $$a!^b$$ b) $$b!^a$$ c) $$a! \cdot b!$$ d) $${2}^{ab}$$ By ...
-1
votes
3answers
52 views

Proving that $\frac{(N+p-1)!}{p!(p-1)!N!}$ is an integer?

Consider the quantity: $$\frac{(N+p-1)!}{p!(p-1)!N!}$$ where $N$ and $p$ are positive integers. How can we show that this is always an integer (which I believe has to be the case since it represents ...
2
votes
1answer
75 views

Is there an equivalent to the Bertrand's postulate between factorials and primorials?

As the title explains, I am trying to know if there is a definition about the upper limit to find the first primorial $p_i\#$ (following the definition at OEIS) existing after a given factorial $n!$ ...
1
vote
4answers
28 views

Prove using factorials that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$

Prove using factorials that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$ I think I'm having a bit of algebra problem with this proof. Here is my work thus ...
1
vote
2answers
47 views

Suppose a coin in tossed $12$ times and there are $3$ heads and $9$ tails. How many sequences…

Suppose a coin is tossed $12$ times and there are $3$ heads and $9$ tails. How many sequences are there in which there are at least $5$ tails in a row? I know this is Permutation with repetition. My ...
0
votes
1answer
33 views

Trying to Express A Factorial As A Polynomial

I'd like to express the following as a polynomial. $$(a-1)(a-2)(a-3) . . . (a-b)$$ where $b<a$ I'm currently working on it now, but wanted to see if anyone's already done it, or already know what ...
1
vote
3answers
47 views

Demonstrating that 1! is = 1

The problem with this explanation is that it's using n = 2 instead of n = 1. Please read the explanation I found on "Math Forum - Ask Dr. Math" ( http://mathforum.org/library/drmath/view/57128.html ). ...
2
votes
2answers
63 views

solve for $\lim_{n \to \infty} \frac{(3n)!(1/27)^n}{(n!)^3}$

I believe the $\lim_{n \to \infty} \frac{(3n)!(1/27)^n}{(n!)^3}$ -> 0. But I am not sure if my reasoning is correct. Because there is a higher power in the denomination that the numerator, the ...
2
votes
2answers
37 views

Help solving the inequality $2^n \leq (n+1)!$, n is integer

I need help solving the following inequality I encountered in the middle of a proof in my calculus I textbook: $2^n \leq (n+1)!$ Where $\mathbf{n}$ in an integer. I've tried applying lg to both ...
3
votes
0answers
38 views

Estimating $n!$ as $e \left(\frac ne \right)^n \le n! \le ne \left(\frac ne \right)^n$

I'm told that for $n \geq 2,$ $$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$ I am then asked to consider $\ln n! = \sum_{k=1}^n \ln k$ and show that for $n \geq 2$ $$n! ...
2
votes
0answers
54 views

Maxima and minima of $f(x)=\binom{16-x}{2x-1}+\binom{20-3x}{4x-5}$.

Find the maximum and minimum value of the function $f(x)=\binom{16-x}{2x-1}+\binom{20-3x}{4x-5}.$ ATTEMPT:- By A.M.-G.M. inequality, $\frac{a+b}{2}\ge\sqrt{ab}$, $\quad$ for $a,b\gt 0$ with equality ...
0
votes
1answer
43 views

solve for $\lim_{n \to \infty} \frac{6(n+1)(6n-1)!}{(6n+5)!}$

I am having trouble solving for this limit with factorials. $$\lim_{n \to \infty} \frac{6(n+1)(6n-1)!}{(6n+5)!}$$ Any hints or suggestions would be great
0
votes
1answer
99 views

finding the smallest number $n$ such that $n!=n(n+1)(n+2)(n+3)$ [closed]

What is the smallest number $n$ such that $n!=n(n+1)(n+2)(n+3)$? How will I solve this type of problems?
0
votes
0answers
22 views

Proving that ${p \choose r}$ is an integer for a prime $p$ and $0 < r < p, r \in \mathbb{Z}$ [duplicate]

I need to prove that given integers $p$ and $r$ such that $p$ is prime and $0 < r < p$, ${p \choose r} = \frac{p!}{r!(p-r)!} \in \mathbb{Z}$ As of now, I don't have any ideas on how to proceed. ...
5
votes
1answer
70 views

The number of zeros in the expansion of $n!$ in base $12$

During an interview last year I was asked the following question: How many zeros appear at the end of $n!$ in base $12$, where $n$ is a positive integer? I applied the known Legendre formula for ...
5
votes
1answer
68 views

Is it true for $n > 2$ then there always exists a prime $\le n$ that does not divide $n$?

I was thinking of how to prove $\frac{n^n}{n!}$ is never an integer for $n > 2$. I think if I prove the above question, then this follows immediately.
1
vote
5answers
60 views

Prove $\sum _{k=2}^{n} k(k-1) {n \choose k}=n(n-1)2^{n-2}$ [duplicate]

Prove $$\sum _{k=2}^{n} k(k-1) {n \choose k}=n(n-1)2^{n-2}$$ Proof by induction: true for $n=2$. Assume true for $n$ and see if $n+1$ is true. $$\sum _{k=2}^{n} k(k-1) {n \choose k}=n(n-1)2^{n-2}$$ ...
5
votes
3answers
209 views

If $x+y+z=3k$, where $x, y, z, k$ are integers, prove that $x!y!z! \geq (k!)^3$

If $x+y+z=3k$, where $x, y, z, k$ are integers, prove that $x!y!z! \geq (k!)^3$ Well I was able to prove this intuitively, but what i need is a rigorous mathematical proof. I shall explain my ...
0
votes
3answers
79 views

Does (9/2)! have a real answer or not? [duplicate]

The TI-84 says 52.342777 but other calculators says domain error.
0
votes
0answers
25 views

$n!>a$, can we solve for $n$ in terms of $a$? [duplicate]

Can we explicitly solve $n$ in terms of $a$? Can we rewrite this inequality in the form of $n>f(a)$ without using $n!$ the factorial?
0
votes
1answer
34 views

Big O for factorials

Hello I have trouble proving:$$(n+1)!\notin O(n!)$$ My first step is the following: $$(n+1)!-cn!\le0$$ Can you please help me with the next step?
1
vote
3answers
55 views

What does an exclamation point raised to a power, with no preceding number, mean?

In the OEIS sequence A049210, I noticed an odd notation I haven't seen before: a(n) = (8*n-1)(!^8), n >= 1, a(0) = 1. What does the ...
-12
votes
3answers
365 views

Is it possible to calculate $\int x! dx$ [closed]

Is it possible to calculate $\int x! dx$, if yes ,then how and if no ,then why not? This question came in my mind when, I solved some questions on integration. Until now I haven't got the right ...
1
vote
1answer
38 views

The exponent of $11$ in the prime factorization of $ 300!$ is___.

The exponent of $11$ in the prime factorization of $ 300!$ is $27$ $28$ $29$ $30$ My attempt: According to Exponent of $p$ in the prime factorization of $n!$ ...
1
vote
2answers
32 views

Combinations equation solving with factorial

I was trying to solve the equation using factorial as shown below but now I'm stuck at this level and need help. $$C(n,3) = 2*C(n,2)$$ $$\frac{n!}{3!(n-3)!} = 2\frac{n!}{2!(n-2)!}$$ $$3! (n - 3)! = ...
1
vote
1answer
37 views

An inequality with exponents, factorials and nth roots!

Problem: Prove for natural numbers $n > 2$, $$(\sqrt{2!}-1)((3!)^{\frac{1}{3}}-\sqrt{2!})\cdots(((n+1)!)^{\frac{1}{n+1}}-(n!)^{\frac{1}{n}}) < \frac{n!}{(n+1)^n}$$. I am unable to do this one. ...
0
votes
0answers
28 views

Modifying permutation function for inputs with equivalent ratios

I have the following function: $$f(a_1,a_2,\ldots,a_n) = \frac{(a_1 + a_2 + \cdots +a_n)!}{(a_1! a_2! \cdots a_n!)}$$ where $a_i\ge 0$ I need to modify this function such that $f(a_1,a_2,...,a_n) = ...
2
votes
5answers
402 views

Divisibility via Induction: $8^n\mid (4n)!$ [duplicate]

I have to prove by induction the following fact: Show that $(4n)!$ is divisible by $8^n$. My stab at the solution: I have a slightly bad feeling about this solution and I would like if someone ...
1
vote
3answers
69 views

Use induction to prove that that $8^{n} | (4n)!$ for all positive integers $n$

Use induction to prove that that $8^{n} | (4n)!$ for all positive integers $n$ So far I have: Base case (n = 1) = $8^{1} | (4(1))!$ = $8 | 24$ which is true. Induction Step: $8^{n + 1} | (4(n + ...
1
vote
2answers
35 views

Congruence Modulo involving factorials

How do I show that $23!\equiv 21! \pmod{101}$? I tried using a calculator but the numbers are so big that am finding it hard to prove. How can factorials be broken down so that they can be easily ...
0
votes
1answer
63 views

Prove that $n! = O(n^n)$

I thought $n^n$ was greater than $n!$. How would I go about proving this? I have this so far: Assume that $P$($n$) is true $n!$ = O($n^n$) Assume that $P$($n+1$) is also true $(n+1)! ...
0
votes
3answers
55 views

Proof by mathematical induction that $2^n < (n+2)!$ for all $n\ge 0$

I have been trying to get this.. For hours. Prove by M.I. that $2^n < (n+2)!$ for $n\ge0$ Here is what I am doing: Base case checks out at $n=0$ Make assumption for: $n=k$ Want to prove: ...
2
votes
2answers
44 views

How to show using proof by induction: $\sqrt[n]{n!} \leqslant \frac{n+1}{2}, n \in \mathbb{Z}^+$

I'm having quite a few problems with the following proof by induction question: $$\sqrt[n]{n!} \leqslant \frac{n+1}{2}, n \in \mathbb{Z}^+$$ I manage to do the easy parts of the base step ($n=1$) ...
0
votes
1answer
46 views

Majoration of the $p$-adic valuation of a factorial.

Let $p$ be a prime number. In order to prove a result on $p$-adic interpolation of iterates, I need to show the following: Lemma. Let $m$ be an integer, one has: ...
0
votes
1answer
45 views

Last digit of a number

I was currently solving a question of permutations and in that I had to find the total ways of something. The answer was ${8\choose 4}$ which has last digit $0$ . A random thought that came to my ...
1
vote
1answer
36 views

“Binary-Like” Function?; In Consecutive Products as Multi-Factorials…

Summary Is there a function $Z(a,b)$ or how would one find such a function so that for $a,b\in \mathbb N$, it would produce $0$'s on for each $a$th step for each $b$th value? For example: $a=2$, ...
-1
votes
1answer
19 views

Fractional numbers' factorial [duplicate]

Is there a law or anyway to know the factorial of a fractional number, because as I see the law of factorization n! = n x (n-1) x (n-2) x ... x 3 x 2 x 1 isn't ...
12
votes
3answers
141 views

Prove $\sum_{q=\alpha}^p \binom{q}{\alpha} \binom{p}{q}\frac{(-1)^q(-q)^p}{q^\alpha}=\frac{p!}{\alpha!}.$

How to prove $\displaystyle \sum_{q=\alpha}^p \binom{q}{\alpha} \binom{p}{q}\frac{(-1)^q(-q)^p}{q^\alpha}=\frac{p!}{\alpha!}$ for $1 \leq \alpha \leq p$? EDIT: This is a result that I derived ...
2
votes
2answers
55 views

How to solve equation with factorial using algebra?

I bring this sample in order to ilustrate $$x! = 2^x + 8$$ I know the answer is $x=4$ but I dunno how to prove it. I mean, if i put the number 4 by observation, tryal and error, I can get the ...
6
votes
1answer
59 views

Combinatorial argument for $1+\sum_{r=1}^{r=n} r\cdot r! = (n+1)!$ [duplicate]

Combinatorial argument for $$1+\sum\limits_{r=1}^{r=n} \ r\cdot r! = (n+1)!$$ The algebraic proof is easy as $r=(r+1)-1$.
0
votes
1answer
19 views

function representation of power series

What is the function representation of this power series? [Summation from n=0 to infinity of ($x^n)(n+1)!/n!$ The solution is $\frac{1}{(1-x)^-2}$ but how??? I know that ...